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%I #15 Apr 18 2016 04:33:41
%S 1,1,2,5,12,31,83,224,615,1708,4777,13455,38110,108428,309714,887666,
%T 2551575,7353423,21240460,61478489,178269670,517784717,1506162369,
%U 4387201004,12795170784,37359689295,109199349181,319493390481,935616592227,2742209152877,8043500169958,23610710680582,69354125493930,203852682699869,599549063015417,1764338532368820
%N G.f. A(x) satisfies: A(x)^3 = A(x^3) / (1 - 3*x).
%C Compare g.f. to: G(x)^2 = G(x^2)/(1 - 2*x) where G(x) is the g.f. of A123916, the EULER transform of A000048.
%H Vaclav Kotesovec, <a href="/A271929/b271929.txt">Table of n, a(n) for n = 1..400</a>
%F The EULER transform of A046211, where A046211(n) is the number of ternary Lyndon words whose digits sum to 1 (or 2) mod 3.
%F a(n) ~ c * 3^n / n^(2/3), where c = 0.1260671867244258410294918... . - _Vaclav Kotesovec_, Apr 18 2016
%e G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 31*x^6 + 83*x^7 + 224*x^8 + 615*x^9 + 1708*x^10 + 4777*x^11 + 13455*x^12 +...
%e where A(x)^3 = A(x^3) / (1 - 3*x).
%e Also, when expressed as the EULER transform of A046211,
%e A(x) = x/( (1-x) * (1-x^2) * (1-x^3)^3 * (1-x^4)^6 * (1-x^5)^16 * (1-x^6)^39 * (1-x^7)^104 * (1-x^8)^270 * (1-x^9)^729 *...* (1-x^n)^A046211(n) *...).
%e RELATED SERIES.
%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 84*x^7 + 252*x^8 + 758*x^9 + 2274*x^10 + 6822*x^11 + 20471*x^12 + 61413*x^13 + 184239*x^14 +...
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3)/(1 - 3*x +x*O(x^n)))^(1/3)); polcoeff(G=A, n)}
%o for(n=1, 50, print1(a(n), ", "))
%Y Cf. A123916.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Apr 17 2016